Embedded newform 13.12.e.a.10.1

• Label: 13.12.e.a.10.1
• Level: $13$
• Weight: $12$
• Character: 13.10
• Analytic conductor: $9.988$
• Analytic rank: $0$
• Dimension: $22$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.98846134727\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			10.1
Character	\(\chi\)	\(=\)	13.10
Dual form			13.12.e.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-74.2981 + 42.8960i) q^{2} +(-148.593 - 257.371i) q^{3} +(2656.14 - 4600.57i) q^{4} +10410.3i q^{5} +(22080.4 + 12748.1i) q^{6} +(18278.8 + 10553.3i) q^{7} +280049. i q^{8} +(44413.7 - 76926.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 242 q^{3} + 8191 q^{4} + 45528 q^{6} + 128496 q^{7} - 322217 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−74.2981	+	42.8960i	−1.64177	+	0.947877i	−0.661568	+	0.749885i	\(0.730109\pi\)
							−0.980204	+	0.197992i	\(0.936558\pi\)
\(3\)	−148.593	−	257.371i	−0.353046	−	0.611494i	0.633735	−	0.773550i	\(-0.281521\pi\)
							−0.986782	+	0.162056i	\(0.948188\pi\)
\(5\)			10410.3i			1.48980i	0.667177	+	0.744899i	\(0.267502\pi\)
							−0.667177	+	0.744899i	\(0.732498\pi\)
\(7\)	18278.8	+	10553.3i	0.411063	+	0.237328i	0.691247	−	0.722619i	\(-0.257062\pi\)
							−0.280183	+	0.959947i	\(0.590395\pi\)
\(11\)	416025.	−	240192.i	0.778861	−	0.449676i	−0.0571656	−	0.998365i	\(-0.518206\pi\)
							0.836026	+	0.548689i	\(0.184873\pi\)
\(13\)	−887852.	+	1.00194e6i	−0.663211	+	0.748432i
							\(17\)	−3.39695e6	+	5.88368e6i	−0.580256	+	1.00503i	0.415193	+	0.909733i	\(0.363714\pi\)
−0.995449	+	0.0952992i	\(0.969619\pi\)
\(19\)	−1.21696e7	−	7.02612e6i	−1.12754	−	0.650985i	−0.184224	−	0.982884i	\(-0.558977\pi\)
							−0.943315	+	0.331899i	\(0.892311\pi\)
\(23\)	−5.23831e6	−	9.07302e6i	−0.169702	−	0.293933i	0.768613	−	0.639714i	\(-0.220947\pi\)
							−0.938315	+	0.345781i	\(0.887614\pi\)
\(29\)	2.67384e7	+	4.63123e7i	0.242073	+	0.419283i	0.961305	−	0.275488i	\(-0.0888393\pi\)
							−0.719232	+	0.694770i	\(0.755506\pi\)
\(31\)			1.43733e8i			0.901711i	0.892597	+	0.450856i	\(0.148881\pi\)
							−0.892597	+	0.450856i	\(0.851119\pi\)
\(37\)	−2.26686e8	+	1.30877e8i	−0.537421	+	0.310280i	−0.744033	−	0.668143i	\(-0.767090\pi\)
							0.206612	+	0.978423i	\(0.433756\pi\)
\(41\)	−1.16026e9	+	6.69878e8i	−1.56403	+	0.902993i	−0.567188	+	0.823589i	\(0.691969\pi\)
							−0.996842	+	0.0794045i	\(0.974698\pi\)
\(43\)	5.01318e8	−	8.68308e8i	0.520040	−	0.900736i	−0.479689	−	0.877439i	\(-0.659250\pi\)
							0.999729	−	0.0232969i	\(-0.00741631\pi\)
\(47\)		−	9.02249e8i		−	0.573837i	−0.957955	−	0.286918i	\(-0.907369\pi\)
							0.957955	−	0.286918i	\(-0.0926309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.e.a.10.1	yes	22
13.2	odd	12		169.12.a.g.1.1		22
13.4	even	6	inner	13.12.e.a.4.1	✓	22
13.11	odd	12		169.12.a.g.1.22		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.e.a.4.1	✓	22	13.4	even	6	inner
13.12.e.a.10.1	yes	22	1.1	even	1	trivial
169.12.a.g.1.1		22	13.2	odd	12
169.12.a.g.1.22		22	13.11	odd	12